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Ouvrages Année : 2013

Nonabelian Jacobian of Projective Surfaces : Geometry and Representation Theory

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Résumé

The Jacobian of a smooth projective curve is undoubtedly one of the most remarkable and beautiful objects in algebraic geometry. This work is an attempt to develop an analogous theory for smooth projective surfaces - a theory of the nonabelian Jacobian of smooth projective surfaces. Just like its classical counterpart, our nonabelian Jacobian relates to vector bundles (of rank 2) on a surface as well as its Hilbert scheme of points. But it also comes equipped with the variation of Hodge-like structures, which produces a sheaf of reductive Lie algebras naturally attached to our Jacobian. This constitutes a nonabelian analogue of the (abelian) Lie algebra structure of the classical Jacobian. This feature naturally relates geometry of surfaces with the representation theory of reductive Lie algebras/groups. This work’s main focus is on providing an in-depth study of various aspects of this relation. It presents a substantial body of evidence that the sheaf of Lie algebras on the nonabelian Jacobian is an efficient tool for using the representation theory to systematically address various algebro-geometric problems. It also shows how to construct new invariants of representation theoretic origin on smooth projective surfaces.

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Dates et versions

hal-03038369 , version 1 (03-12-2020)

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Igor Reider. Nonabelian Jacobian of Projective Surfaces : Geometry and Representation Theory. Springer Berlin Heidelberg, 2072, pp.216, 2013, Lecture Notes in Mathematics ; 2072, 978-3-642-35661-2. ⟨10.1007/978-3-642-35662-9⟩. ⟨hal-03038369⟩
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